The Levelwise Finite Generation of Free Tambara Functors

TL;DR

This paper proves the finite generation of free polynomial G-Tambara functors in specific cases, expanding understanding of their algebraic structure and stability under certain operations.

Contribution

It establishes the levelwise finite generation of free polynomial G-Tambara functors for particular classes of groups, and shows finiteness conditions are preserved under key operations.

Findings

Finite generation proven for G as a finite Dedekind group.

Finite generation proven for G isomorphic to C_p ⋉ C_q, p > q primes.

Finiteness conditions are stable under box products and norms of Tambara functors.

Abstract

We prove the levelwise finite generation of free polynomial $G$-Tambara functors in a collection of cases, most notably when $G$ is a finite Dedekind group or when $G \cong C_p \rtimes C_q$, $p > q$ primes. In the process, we establish the permanence of various finiteness conditions under box products and norms $n_H^G$ of Tambara functors, including a weak Hilbert Basis Theorem.